The Effect of Aluminum Deformation Conditions on Microhardness and Indentation Size Effect Characteristics

Abstract

The degree and speed of deformation are factors that influence microstructure and mechanical properties. Aluminum (99.5%) was used as the test material in this experiment. This material is currently mainly used in the electrical industry to manufacture conductors as a substitute for the more expensive copper. The cylindrical samples were deformed at a strain rate of up to 2500 s⁻¹, and the degree of deformation was up to 85%. At the point place of maximum deformation, usually in the center of the sample, the microhardness was measured under various loads, between 10 gf and 100 gf. The obtained data were used to determine the characteristics or parameters of the indentation size effect (ISE) and the influence of the deformation conditions on the microhardness. The results obtained were processed by linear regression analysis, followed by the creation of deformation maps.

1. Introduction

The measurement of microhardness has many advantages for materials engineers: it provides fundamental information about the mechanical properties of materials with minimal material input and usually with minimal surface damage. On the other hand, the obtained results are not unambiguous and require a careful approach. The most important factor influencing the measured values is probably the ISE effect. It is essentially dependent on the test load. The geometric similarity of the indentations in the Vickers hardness measurement theoretically ensures that the measured value is independent of the test load.

Three cases can occur with the Vickers method, which is frequently used in microhardness measurement. The first is ideal if the measured value is not influenced by the applied load and the value of Meyer’s index n = 2 (size of the ISE). In this case, we measure the “true value” of the hardness. In the second case of a “normal” ISE (type or characteristic of the ISE), the measured value decreases with increasing test load and n < 2. In the third case of a “reverse” (RISE), the measured value increases with increasing test load and n > 2. The variability of the test load during the measurement cannot be avoided. The higher it is, the greater the indentation and the more accurate the measurement. The greater the indentation, the more we can neglect the effect of the load. The greater the indentation, the more visible the surface and the more it loses value. This is not appropriate for end products. When measuring the hardness of thin layers, small components, and individual phases in an alloy, however small loads are more suitable. It should be noted that the values of the applied load must lie within the interval specified by the relevant standard [1].

The causes of the ISE are related to many factors, such as the properties of the tested material, the properties of the hardness used tester and its parts (especially the indenter), the test conditions, and the preparation of the test surface. The influence of these factors has been analyzed in detail in the works of Tabor [2], Sangwal [3,4], Gong [5], Ren [6], Navrátil [7], and others.

After researching the ISE in metals in their natural, undeformed state [8], a collective of authors focused on the influence of deformation on the magnitude and nature of Meyer’s index, as well as on other characteristics or parameters describing the properties of this phenomenon. The samples were deformed in tension and compression. The results were published, in the paper [9], among others, in which the test methodology, the method for calculating the Meyer’s index and other parameters, and the properties of the used material are described in detail.

In general, the variability of Meyer’s index shows the tendency shown in Figure 1 for tensile deformation with an average strain rate of ε_Lc = 0.0004 s⁻¹ with a reduction in area (contraction) Z of up to 84.5% and for compressive deformation with a strain rate of 0.1–0.01 s⁻¹ with a degree of deformation ε between 5.8 and 83.5% (source [9]). In further investigations, the question of whether a possible increase or decrease in the strain rate has an effect on the monitored ISE parameters was investigated.

One of the results was an attempt to construct a deformation map. The authors based their work on the construction of deformation maps for titanium alloys (α + β alloys VT3-1, VT8; α alloy OT4). The deformation rate (or degree of deformation) was plotted on one axis and the forming temperature on the other. In the map, areas with a homogeneous globular/lamellar structure (morphology of the α phase) up to areas with cracks were marked. In titanium alloys, the situation is complicated by polymorphic transformations in the forming temperature range and the generation of deformation heat. Another anomaly is the high inhomogeneity of the deformation as a result of the hexagonal crystallographic orientation at a temperature below the polymorphic transformation temperature. These tests had to be carried out at elevated temperatures. As the research focused on a “special” production, the results, or fragments, e.g., [10,11,12], were unfortunately not published.

Let us return to aluminum. The strain rate mentioned above is not high. The question arises as to how a higher or lower strain rate affects the monitored parameters and whether it is statistically significant. In contrast to titanium inspiration, aluminum can also be deformed at ambient temperature (during the test it fluctuated in the range of 19.0–28.4 °C (

Table 1. Parameters of compression: the degree (amount) of the deformation ε (%), strain rate $\dot{\phi }$ (s⁻¹), values of the microhardness and average microhardness ($\overline{HV}$), the ambient temperature of deformation, results of calibration, and measured and true microhardness H_PSRA₁.

	Tester	ε (%)	$\dot{\mathit{\phi }}$ (s−1)	ln ($\dot{\mathit{\phi }}$)	T (°C)	Calibration			Microhardness
						rrel (%)	Erel (%)	Urel (%)	HV	HV0.01	HV0.025	HV0.05	HV0.1	HPSRA1
1	DWT	68.57	568.555	6.343	23.9	2.49	−1.14	6.05	51	51	55	50	50	49
2	DWT	62.64	479.265	6.172	23.9	2.49	−1.14	6.05	49	48	52	48	47	47
3	DWT	65.28	521.244	6.256	23.9	2.49	−1.14	6.05	51	51	53	50	49	49
4	DWT	29.73	133.093	4.891	22.7	3.07	−1.82	6.98	40	38	41	40	41	41
5	DWT	67.41	600.643	6.398	20.3	2.19	−2.35	7.20	52	54	51	50	54	54
6	DWT	80.50	1506.624	7.318	20.3	2.19	−2.35	7.20	50	51	49	49	50	50
7	DWT	84.62	2506.770	7.827	22.7	3.21	−1.99	7.14	53	56	52	51	52	52
8	HLR	69.55	0.051	−2.981	28.0	3.1	3.71	8.46	43	31	44	49	48	50
9	HLR	50.55	0.030	−3.504	28.0	3.1	3.71	8.46	40	29	44	43	44	46
10	HLR	58.43	0.036	−3.323	28.0	3.1	3.71	8.46	44	32	44	50	49	51
11	HLR	16.61	0.028	−3.560	28.0	3.1	3.71	8.46	38	33	37	35	45	46
12	WPM	63.39	0.130	−2.043	28.0	3.07	−1.82	6.98	35	36	43	47	48	50
13	WPM	69.87	0.120	−2.119	28.0	3.1	3.71	8.46	41	30	42	45	45	47
14	WPM	43.53	0.080	−2.522	28.0	3.1	3.71	8.46	38	31	37	42	42	44
15	WPM	9.15	0.025	−3.682	28.4	3.1	3.71	8.46	28	25	28	28	29	30
16	WPM	10.64	0.029	−3.551	28.4	3.1	3.71	8.46	29	27	31	30	30	30
17	WPM	20.48	0.059	−2.838	28.4	3.1	3.71	8.46	32	28	34	35	32	32
18	WPM	31.02	0.054	−2.916	28.4	3.1	3.71	8.46	36	32	37	39	37	37
19	WPM	54.33	0.125	−2.078	26.6	3.47	4.63	9.67	38	29	41	41	41	43
20	WPM	54.01	0.095	−2.357	28.4	3.1	3.71	8.46	43	34	46	47	45	46
21	WPM	50.40	0.114	−2.170	28.1	3.47	4.63	9.67	40	31	41	44	44	46
22	WPM	63.08	0.112	−2.192	28.1	3.47	4.63	9.67	42	34	42	45	46	48
23	WPM	45.15	0.091	−2.392	29.9	3.47	4.63	9.67	36	29	37	38	38	39
24	ZWICK	5.80	0.029	−3.536	19.0	7.34	4.91	9.28	30	30	32	30	29	29
25	ZWICK	16.20	0.038	−3.269	19.0	7.34	4.91	9.28	34	32	35	33	35	35
26	ZWICK	27.20	0.064	−2.748	19.5	2.16	0.85	5.62	37	35	37	39	39	39
27	ZWICK	56.10	0.056	−2.880	19.5	2.16	0.85	5.62	44	38	45	45	46	47
28	ZWICK	65.40	0.070	−2.659	19.0	7.34	4.91	9.28	43	37	44	48	45	46
29	ZWICK	73.10	0.049	−3.012	20.0	2.16	0.85	5.62	43	38	43	46	47	48
30	ZWICK	75.10	0.094	−2.366	19.0	2.16	0.85	5.62	44	37	45	48	46	47
31	ZWICK	76.50	0.092	−2.383	19.5	2.16	0.85	5.62	45	38	47	49	48	49

). The degree of deformation was chosen in the range of up to 99.9%, and the resulting value was usually a few percent higher or lower than the chosen value, probably due to the elasticity of the tested material, the inertia of the device, etc. Due to the greater complexity of tensile tests (production samples), we have limited ourselves to compression tests.

The aim of this work is to investigate the influence of different forming conditions of aluminum at ambient temperature (i.e., without heating) on the microhardness and ISE parameters. The obtained dependencies are evaluated graphically and with the help of Design of Experiments (DoE) or used to create a deformation map.

2. Materials and Methods

The test material was commercially available pure aluminum (99.5% Al according to the standards ČSN 42 4004 [13] which corresponds to aluminum EN AW 1350/E-Al99.5 according to the Aluminum Association (AA) [14]), whose mechanical and other properties are described in more detail in [9]. The semi-finished product was annealed at 400 °C/1 h → cooled in the furnace. Cylindrical samples with a height of 20 mm and a diameter of 9 mm were turned. No lubricant was used during deformation. All deformations were carried out at room temperature (Table 1).

An HLR hydraulic press (type 12, manufacturer Proma, Sezemice, Czech Republic) and a WPM testing device (type ZDM30T, manufacturer VEB Werkstoff Prüfmachinen, Leipzig, Germany) were used as testing devices that generate lower strain rates. A Zwick tester (manufacturer Zwick GmbH & Co., Ulm, Germany) was used to determine the average strain rate. High strain rates were generated with a drop weight tester (DWT) as a function of the drop height (maximum 3 m). The mass of the ram was 70 kg, its velocity was calculated as free-fall velocity with a local acceleration due to gravity g = 9.8092 s⁻² [15], and friction was neglected.

The deformed cylinders were cut with a cooled diamond saw parallel to the axis at maximum diameter. The metallographic surface was polished on papers in the order 80 … 3000 ANSI/CAMI and polished with diamond paste (last fraction grain size 0.5 µm). The surface was then etched with 0.7% aqueous HF solution to visualize the material flow during deformation. The microhardness was measured at the point of maximum deformation, usually in the center of the surface (X-shaped area or the “forging cross”). All measurements were performed by an operator using a Hanemann handheld tester (type Mod D32, part of the optical microscope Neophot-32, manufacturer Carl Zeiss, Jena, Germany) with a magnification of 480×. Test loads of 10, 25, 50, and 100 gf with a load duration of 15 s were used.

Due to the large number of samples, the microhardness measurements had to be carried out over several days. Before each measurement, the tester was calibrated according to ISO 6507-2 [1] at a load of 50 gf. The results of the repeated calibrations, which met the requirements of the standard (relative repeatability r_rel, tester error E_rel, and expanded calibration uncertainty U_rel) are shown in Table 1. For each load, five indentations were placed in random order so that the group of indentations was at the point of maximum deformation, and at the same time, the distances between the indentations were minimized, fulfilling the requirements of ISO 6507-1, 2 [1,16]. The forming conditions and the resulting microhardness values are listed in Table 1.

Based on a one-way ANOVA with significance level α = 0.05, it can be said that the strain rate has no statistically significant effect on the microhardness HV0.05 (p = 0.9891, the variability of the microhardness due to the change in strain rate can only be explained by α = 0.7%) and also has no statistically significant effect on the type and size of ISE, expressed by the Meyer’s index n (p = 0.5962, α = 10.8%, which is a slightly larger effect than for the microhardness).

The representativeness of the above data is somewhat diminished by the fact that the experiment was not balanced (balanced experimental design), but average values (strain rate) and four selected values of the degree of deformation (over 50%) were used for each test device.

What about the load? The influence of the test load on the measured hardness value is statistically significant, as p = 0.000419.

3. Results

The methodology and procedures for determining the basic parameters that determine the nature and size of the ISE calculation was based on works [8,9], whereby works [4,7,17,18,19,20] was also used. Meyer’s power law, proportional sample resistance (PSR), and the Hays–Kendall approach were most commonly used to determine the ISE characteristics.

If the value of Meyer’s index n is 2 (±0.05), the test load has no influence on the measured microhardness value and Kick’s law applies. If it is lower, it is a “normal” ISE, typical for brittle materials, such as ceramics, semiconductors, and sintered materials. Higher values, on the other hand are typical of a “reverse” ISE (RISE), which is normally associated with plastic materials, such as pure metals. With increasing deformation, a hardening occurs in metallic materials that exhausts the possibilities of plastic deformation, and in extreme cases, the metal behaves like a brittle material with a “normal” ISE.

The other parameters that characterize the ISE include the value c₁ (N mm⁻¹), which characterizes the elastic properties of the material; c₂ (N mm⁻²), which characterizes the plastic properties of the material; and c₀ (N), which is a measure of the residual surface stress. The parameter c₁ characterizes the load dependence of the microhardness and describes the ISE in the PSR model. It consists of the elastic resistance of the test specimen between the indenter and the sample. The ratio c₁/c₂ (mm) is the measure of the residual stress, the result of mechanical processing, especially the grinding and polishing of the sample. The parameter W, which is related to elasticity and also to deformation strengthening, represents the minimum load at which a visible indentation occurs.

When evaluating the influence of residual stress, the result of mechanical processing (c₁/c₂), the authors found that the samples from which the metallurgical surface was produced (they were cast together and therefore processed at the same time) were produced under the same conditions (A–H in Figure 2). For some samples, particularly series B, H, E, and F, the results were also influenced by other factors such as the degree and speed of deformation. Their identification will be the subject of further investigations.

Parameter A₁ (other parameters can also be used, e.g., c₂), can be used to calculate the “true hardness”, which is not influenced by the applied load. The parameter A₁ (N mm⁻²) is not dependent on the load, and therefore the “true hardness” HPSRA₁ calculated with it is not dependent on the applied load. The parameter is dependent on the parameter W (N; the unit gf was used in this paper for clarification), which characterizes the elasticity and deformation resistance of the tested material. In practice, this is the minimum test load that causes a visible impression during the hardness test.

The values of the Meyer’s index and other ISE parameters were calculated on the basis of Table 1 and are listed in

Table 2. ISE parameters.

	Tester	n	Amoc	Aln	c0 (N)	c1 (N mm−1)	c2 (N mm−2)	W (N)	W (gf)	A1 (N mm−2)	c1/c2 (mm)
1	DROP WEIGHT	1.9620	238.50	5.474	−0.004	0.887	249.3	0.010	1.10	260.1	0.0036
2	DROP WEIGHT	1.9690	232.40	5.448	−0.016	1.560	229.3	0.011	1.15	247.7	0.0068
3	DROP WEIGHT	1.9700	243.50	5.495	−0.032	2.563	227.5	0.012	1.28	258.3	0.0113
4	DROP WEIGHT	2.0405	239.94	5.480	0.016	−1.124	228.19	−0.007	−0.69	216.01	−0.0049
5	DROP WEIGHT	1.9840	260.90	5.564	0.089	−5.969	358.3	−0.012	−1.24	283.3	−0.0167
6	DROP WEIGHT	1.9750	241.70	5.487	0.040	−2.477	293.8	−0.003	−0.36	263.8	−0.0084
7	DROP WEIGHT	1.9380	226.10	5.421	0.048	−2.757	308.1	0.001	0.10	273.8	−0.0089
8	HLR	2.4338	920.90	6.825	−0.240	10.044	153.33	−0.040	−4.09	265.9	0.0655
9	HLR	2.3723	687.10	6.533	−0.165	6.367	171.96	−0.034	−3.50	240.71	0.0370
10	HLR	2.4258	916.33	6.820	−0.215	8.808	171.35	−0.042	−4.23	271.15	0.0514
11	HLR	2.2463	433.45	6.072	0.345	−21.154	482.24	−0.070	−7.15	244.79	−0.0439
12	WPM	2.2959	602.07	6.400	−0.072	1.875	242.14	−0.036	−3.64	263.56	0.0077
13	WPM	2.4022	777.32	6.656	−0.180	6.867	175.36	−0.040	−4.10	250.5	0.0392
14	WPM	2.3110	541.87	6.295	−0.084	2.192	209.34	−0.039	−3.95	232.77	0.0105
15	WPM	2.1295	215.88	5.375	0.008	−1.114	166.79	−0.019	−1.94	156.67	−0.0067
16	WPM	2.0790	198.56	5.291	−0.018	0.543	155.09	−0.005	−0.51	160.13	0.0035
17	WPM	2.1255	251.25	5.526	−0.157	7.161	103.23	0.006	0.61	171.31	0.0694
18	WPM	2.1300	289.81	5.669	−0.110	5.024	146.79	−0.003	−0.29	198.03	0.0342
19	WPM	2.3136	542.50	6.296	−0.121	4.398	179.78	−0.030	−3.00	226.3	0.0245
20	WPM	2.2482	508.77	6.232	−0.179	8.441	150.68	−0.012	−1.26	244.49	0.0560
21	WPM	2.3364	622.82	6.434	−0.122	4.287	196.6	−0.035	−3.60	243.38	0.0218
22	WPM	2.2964	579.71	6.363	−0.070	1.737	235.29	−0.036	−3.66	254.81	0.0074
23	WPM	2.2340	392.56	5.973	−0.126	5.061	152.38	−0.017	−1.73	203.85	0.0332
24	ZWICK	1.9840	151.50	5.020	−0.054	2.938	125.6	0.013	1.39	152.7	0.0234
25	ZWICK	2.0710	221.60	5.401	0.060	−3.547	222.1	−0.015	−1.56	186.5	−0.0160
26	ZWICK	2.1050	275.40	5.618	−0.019	0.261	205.5	−0.013	−1.30	208.3	0.0013
27	ZWICK	2.1490	374.30	5.925	−0.023	0.392	241.6	−0.016	−1.64	246.1	0.0016
28	ZWICK	2.2000	440.10	6.087	−0.146	7.004	164.8	−0.009	−0.98	242.9	0.0425
29	ZWICK	2.1920	427.70	6.058	−0.025	−0.042	253	−0.026	2.65	252.5	−0.0002
30	ZWICK	2.2096	461.34	6.134	−0.121	5.518	188.1	−0.015	−1.47	250.52	0.0293
31	ZWICK	2.2200	491.60	6.197	−0.110	4.924	202.2	−0.016	−1.73	258.7	0.0244

.

Regarding the correlation between the parameters listed in Table 2, the unpaired t-test revealed a strong correlation between c₀ and c₁ (r² = −0.9807), c₀ and the ratio c₁/c₂ (r² = −0.9403; in this case, the expected linear relationship was confirmed with a coefficient of determination R² in the range of 0.9279 for Zwick and 0.9830 for HLR), and c₁ and the ratio c₁/c₂ (r² = −0.9588). There is a large correlation between n and c₀, c₁, the ratio c₁/c₂, and W, as well as between c₂ and the ratio c₁/c₂ (r² = between 0.7 and 0.9).

To determine the statistical significance of the differences between the measured values of HV0.05 and the calculated values of Meyer’s index n, obtained after deformation on individual test specimens, an unpaired t-test with a significance level of α = 0.05 was performed.

As shown in

Table 3. The two-tailed p value—statistical significance of the difference in the hardness HV0.05 of samples pressed by individual testers.

	Zwick	WPM	DWT
HLR	0.6387	0.2954	0.2511
DWT	0.0742	0.0093
WPM	0.5256

, although the difference in hardness between samples pressed on different testers is not statistically significant, it can be seen that at a higher strain rate achieved by forming on the drop weight tester (DWT), this difference is greater than between samples pressed on the HLR, WPM, and ZWICK testers at a lower strain rate.

For Meyer’s index n, the difference between the specimens deformed by individual testing devices is considered statistically significant in all cases. The differences between specimens molded at higher strain rates per drop weight testing machine and lower strain rates on the HLR, WPM and ZWICK presses are highly statistically significant, as shown in

Table 4. The two-tailed p value—statistical significance of the difference in Meyer’s index n of samples pressed by individual testers.

	Zwick	WPM	DWT
HLR	0.0013	0.0429	0.0001
DWT	0.0001	0.0001
WPM	0.0332

.

As can be seen in Figure 3, the microhardness HV0.05 increases with the degree of deformation ε, with no significant effect of the strain rate. Similarly, the microhardness increases with the increase in the strain rate $\dot{\phi }$, which is more significant at lower strain rates. For samples deformed at maximum speed (DWT tester), the hardness also increases with the increase in strain rate $\dot{\phi }$; this increase is slower, as can be seen in Figure 4.

At lower degrees of deformation ε and strain rates $\dot{\phi }$, the value of Meyer’s index n gradually increases from an approximately neutral ISE to a RISE, essentially as in Figure 1. For samples deformed at maximum rate, the trend is different. As ε and $\dot{\phi }$ increase, Meyer’s index n decreases from a neutral value to a slightly “normal” value, as in less plastic (brittle) materials. The possibilities of plastic deformation are exhausted more quickly at higher strain rates than at lower ones, as shown in Figure 5 and Figure 6.

As with the individual ISE parameters, the surface residual stress c₀ decreases slightly with increasing ε and increases slightly with increasing $\dot{\phi }$, regardless of the degree of deformation applied ε. It is advisable to use other methods for measuring c₀ (e.g., incremental drilling, ultrasound, and diffraction methods), which are described in [21,22,23], for example, and to compare the results with the values determined using ISE analysis.

The parameters c₁, c₂, and c₁/c₂ increase slightly with increasing ε and also $\dot{\phi }$ for slower deformations, and they decrease slightly for fast deformations of the DWT. The parameter W behaves chaotically. As already mentioned in [9], for example, the mentioned parameter is subject to several anomalies and deserves a more detailed investigation.

For the compression set test, a multiple linear regression (EXCEL → LINEST program) was used to investigate the simultaneous effect of several factors on the microhardness values and ISE parameters. The value of the coefficient of determination r² = 0.3944 for HV0.05 and r² = 0.6875 for n. They indicate a large correlation for the Meyer’s index, but only a medium correlation (HV0.05). This means that 39.4% (68.7%) of the variation in the microhardness or n can be explained by the effect of these factors [24,25]. It follows that the results obtained with this method are quite meaningful. In further studies, it will be necessary to supplement the tests with experiments at deformation rates lower than those achieved using DWT to obtain a more uniform database of input values.

The result of the regression is a linear Equation (1) in the following form:

$$HV (or ISE parameter) = \mathrm{d}+(a\ast \epsilon )+(b\ast ln(\left(\dot{\phi }\right)\left)\right)$$

(1)

This equation was used after inserting the real values to create deformation maps, e.g., for HV0.05 in Figure 7 and for Meyer’s index n in Figure 8.

4. Deformation Maps

One of the pioneers in the creation of deformation maps was M.F Ashby, who published an article [26] as early as 1972 in which he explained the basic areas of use of the application of deformation maps. The maps primarily enable the study of crystal structure and atomic bonding in plastic flow, help in planning experiments to study a particular mechanism, and are useful in selecting a material for engineering applications, predicting its deformation mechanism and strain hardening mechanisms. The aim of the maps is to illustrate the way in which alternative deformation mechanisms “compete” by creating maps in stress/temperature space. The space is divided into fields. Within a field, one mechanism is dominant. An example of the use of deformation maps is the work of Sargent and Ashby [27] from 1982. They created deformation maps for a group of metals with similar properties using coordinates:

(a)

Temperature or homologous temperature (x-axis) and normalized shear stress or shear stress at 20 °C (y-axis);

(b)

Normalized shear stress/shear stress at 20 °C (x-axis) and shear strain rate (y-axis).

This work was of great practical importance, as it examined a group of metals (Ti, Zr, and Hf) that are becoming increasingly important in technical practice. Ti and its alloys are increasingly used in aerospace and chemical engineering. Zr and alloys based on it have important structural applications in certain nuclear reactors. Hf can replace Zr in the production of refractory alloys. However, it is not yet produced in large quantities and is therefore still awaiting application. Many applications of Ti and Zr require the designer to have a good understanding of low-temperature plasticity and high-temperature strength. The maps in this paper were created using the method and equations of Frost and Ashby [28].

The Compendium of deformation-mechanism maps for metals [29] shows that the use of deformation maps is still justified. The maps have axes indicating either the normalized stress σ/G (where G is the shear modulus) or the homologous temperature T/Tm (where Tm is the melting point of the material).

There are representations of processing conditions, usually in the terms of strain and temperature. A typical example is the work by Prasad et al. from 2015 [30]. While such strain–temperature diagrams are useful, they cannot capture the more fundamental aspects of deformation, as there are many variables that vary from material to material.

Deformation maps, whose construction is described by Mohamed and Langdon, 1974 [31], show either the stress σ or the normalized stress σ/G (or the normalized stress), where G is the shear modulus, as a function of the temperature, T, in degrees Kelvin or the homologous temperature, T/Tm (where Tm is the melting point of the material). Using the best governing/constitutive equations available to describe each of the mechanisms that many occur during steady-state flow, a custom map is created. The map is divided into fields, with each field dominated by a particular mechanism. Such maps are increasingly used in practice (e.g., when describing the behavior of fuel in a reactor). The disadvantage of this form of deformation mechanism map is that it must be created for a specific grain size. The authors have used the behavior of pure aluminum as an example for the creation of such a map.

The question of deformation maps and their practical application is still dealt with today, as can be seen from the work [32], which deals with the creation of a map of the hot deformation of the titanium alloy Ta 15.

As can be seen in Figure 9 (HV 0.05) and Figure 10 (Meyer’s index n), the deformation maps obtained by linear regression are simple and not very meaningful. Therefore, the Origin 8 program was used as an additional tool. With its help, the deformation maps in Figure 7 (HV0.05), Figure 8 (Meyer’s index n), Figure 11 (“true hardness” H_PSRA₁), Figure 12 (c₀), Figure 13 (W), and Figure 14 (ratio c₁/c₂) were created.

Figure 7. Deformation map for HV0.05 as a function of deformation ε and strain rate $\dot{\phi }$ created by Origin 8.

Figure 7. Deformation map for HV0.05 as a function of deformation ε and strain rate $\dot{\phi }$ created by Origin 8.

Figure 8. Deformation map for Meyer’s index n as a function of deformation ε and strain rate $\dot{\phi }$ created by Origin 8.

Figure 8. Deformation map for Meyer’s index n as a function of deformation ε and strain rate $\dot{\phi }$ created by Origin 8.

Figure 9. Deformation map for HV0.05 as a function of deformation ε and strain rate $\dot{\phi }$ created by regression.

Figure 9. Deformation map for HV0.05 as a function of deformation ε and strain rate $\dot{\phi }$ created by regression.

Figure 10. Deformation map for Meyer’s index n as a function of deformation ε and strain rate $\dot{\phi }$ created by regression.

Figure 10. Deformation map for Meyer’s index n as a function of deformation ε and strain rate $\dot{\phi }$ created by regression.

Figure 11. Deformation map for “true hardness” H_PSRA₁ as a function of deformation ε and strain rate $\dot{\phi }$ created by Origin 8.

Figure 11. Deformation map for “true hardness” H_PSRA₁ as a function of deformation ε and strain rate $\dot{\phi }$ created by Origin 8.

Figure 12. Deformation map for c₀ as a function of deformation ε and strain rate $\dot{\phi }$ created by Origin 8.

Figure 12. Deformation map for c₀ as a function of deformation ε and strain rate $\dot{\phi }$ created by Origin 8.

Figure 13. Deformation map for W as a function of deformation ε and strain rate $\dot{\phi }$ created by Origin 8.

Figure 13. Deformation map for W as a function of deformation ε and strain rate $\dot{\phi }$ created by Origin 8.

Figure 14. Deformation map for the ratio c₁/c₂ as a function of deformation ε and strain rate $\dot{\phi }$ created by Origin 8.

Figure 14. Deformation map for the ratio c₁/c₂ as a function of deformation ε and strain rate $\dot{\phi }$ created by Origin 8.

These maps are better structured than the maps created using linear regression and provide more information.

For microhardness and Meyer’s index n, the DoE (Design of Experiments) method was used as an additional method for creating deformation maps. The DoE method was used using the help of Quantum XL software (2016 v5.60) to create a model to predict microhardness and Meyer’s index n [33]. The DoE method allows the manipulation of multiple input factors and the monitoring of their effects on specific outcomes (responses). Based on experimental data and using the DoE method of historical analysis, a model was created. The experimental matrix consists of two factors with the different values mentioned above, the degree of deformation ε and the deformation $\dot{\phi }$, while one output (HV0.05 or n) was monitored [34,35,36]. Examples of deformation maps created using DoE can be seen in Figure 15 (for “true hardness” H_PSRA₁) and Figure 16 (for Meyer’s index).

5. Discussion

The simultaneous influence of deformation and strain rate on HV0.05 in the map was determined using the program. In contrast to the map obtained by regression (Figure 7), Origin 8 (Figure 9) is not unambiguous. The hardness increases with the growth of the two factors. Paradoxically, its value is also higher in the range ε = 15–30% and ln ($\dot{\phi }$) in the interval −2 to +2 s⁻¹. This is a range that was extrapolated by the program. It may therefore be an insufficient database (lack of measured values in a certain interval of strain rates, which has already been mentioned above) or a program error, or these values may correspond to reality. This represents a challenge for the authors: to find a suitable device that allows the deformation of specimens in a given interval and the completion of the input data. The value of the “true hardness” H_PSRA₁ shows a similar tendency with a certain anomaly (higher hardness) in the low-strain-rate range, which applies to the maps created with the Origin 8 and DoE programs. The influence of the two factors on the Meyer’s index is rather contradictory. The tendency from an inverted ISE (n > 2), which is most pronounced in the interval of the range ε = 40–70% and ln ($\dot{\phi }$) in the interval −3 to +2 s⁻¹, is towards a neutral to “normal” ISE in the direction of a lower degree of deformation and a higher deformation rate. Thus, an increase in the strain rate counteracts an increase in the degree of deformation as a factor that shifts the ISE into the opposite range.

Due to the differences found in the investigation, the authors will focus on the evaluation of those parameters influencing the ISE, c₀, c₁/c₂, and W. It will also be necessary to supplement the input data to evenly cover the intended strain rate range.

The issue of the ISE is still an object of scientific research. A certain problem is that research in this area is focused on materials other than metals. And it is precisely metals, which are expected to have a tendency for plastic deformation and therefore to reverse the ISE, that are the object of our interest as metallurgists–materialists. As we mentioned in previous contributions (ISE of metals, ISE of deformed copper and aluminum), the reverse ISE is given less attention in the professional literature, which significantly narrows the possibilities of citations, especially current citations from this area. The same is true of works that describe the interaction of ISE parameters with the influence of the monitored material by deformation, radiation, temperature, etc. Despite the above limitations, the contribution was supplemented, on the advice of the editors, with up-to-date citations, which expanded the discussion of the results.

Among the factors that are monitored in the professional literature in connection with the ISE as influencing the ISE, radiation effects, chemical–thermal processing, entropy, surface layers, crystallographic orientations, indentation strain rates, the grain size of the tested material, and, only to a small extent, applied deformation dominate.

The study of Xia et al. [37] aimed to investigate the microstructure information and mechanical behavior of ion-irradiated zirconium (Zr) alloys; their mechanical properties were characterized by nanoindentation tests, which revealed a significant indentation size effect (ISE) and irradiation hardening behavior. The results indicated that the ISE phenomenon could be attributed to the reduction in the density of geometrically necessary dislocations (GNDs) caused by the expansion of the plastic zone. Lai et al. [38] studied ion irradiation combined with nanoindentation with a Berkovich indenter. Two well-characterized RPV steels, each ion-irradiated to up to two different levels of displacement damage, were investigated. The measured hardening profiles were compared with predictions based on different DBH (the dispersed barrier hardening) models. These include the role of the unirradiated microstructure, the proper treatment of the indentation size effect (ISE), and the appropriate superposition rule of individual hardening contributions. Zhang et al. [39] describe a model to assess the nanoindentation hardness of shallow ion-irradiated materials, built upon two factors: (1) the contribution of the damage layer to the indentation size effect (ISE) becomes negligible when the indentation depth surpasses a certain value; (2) the difference in the profile of the hardness between non- and ion-irradiated samples can still be approximated as a coated system. Comparing the fitting results from the Nix–Gao, Korsunsky, modified NGK, and new models reveals that without introducing additional parameters (while maintaining the three degrees of freedom of the Korsunsky model), the new model effectively describes the shallow ion irradiation hardness data of various materials. In the experiment, seven alloys were used: V–4Cr–4Ti prepared using vacuum arc remelting and by powder metallurgy, pure Fe, CLAM steel, 15Cr ODS–steel, and a FeCrV high-entropy alloy. Xi et al. [40] also were dealing with the effect of He ion irradiation dose on the surface and mechanical properties of Al–Mg–Sc–Zr alloys modified with 0.5 wt% multi-walled carbon nanotubes (MWCNTs) manufactured by laser powder bed fusion and its influence on microhardness. Luo and Kitchen [41] dealt with the ISE and true hardness of plastically deformed Hadfield steel. Samples of different plastic straining conditions were tested by the Vickers microhardness method, using a range of loads from 10 to 1000 g. The results obey Meyer’s power law with n < 2. The plastically strained samples showed not only significant work hardening but also different ISE significance, as compared to the non-deformed bulk steel.

The influence of indentation strain rates on the ISE has been studied by several authors. Among the earlier published papers, we can mention Ma et al. [42], who studied this phenomenon on a titanium alloy. Among the more recent ones, Shrestha et al. [43] studied the relationship between the ISE, strain rate, and ductility in soda–lime silica glass by Vickers method with applied loads of 12.5, 25, 35, 50, 60, 75, and 100 grf. The ISE, while more pronounced at slower strain rates, exhibited a diminishing influence with increases in the strain rate for constant loading rate (CLR) tests whereas little influence from strain rate was observed for the constant strain rate (CSR) tests. A paper published by Asumadu et al. [44] explores the effects of surface carbo-nitriding on the material length scale and dislocation microstructure mechanisms of 1045 steel by microstructural and mechanical (micro/nano-indentation) characterization, including ISE study. The microhardeness was tested by the Vickers method with a conical diamond indentor with loads of 0.3, 1, 3, 10, 30, and 50 N.

Petruš et al. [45] investigated the indentation load–size effect of high-entropy carbides with different applied indentation loads from 50 mN to 10 N (Vickers and Berkovich methods) on (Hf–Ta–Zr–Nb–Ti) C and (Mo–Nb–Ta–V–W) C high-entropy carbides. The load dependence of hardness was analyzed using the traditional Meyer’s law, the proportional specimen resistance model, and the modified PSR (proportional specimen resistance) model. The values of Meyer’s index n ranged between 1.864 and 1.895.

The nanohardness and ISE of high-entropy Ti–Nb–Zr–Ta–Mo alloys exhibiting dendrite microstructures was investigated by Aranda et al. [46]. The presence of an ISE was detected at low loads. The dendritic and interdendritic phases exhibited an inverse ISE for alloys without Mo. In contrast, in alloys containing Mo, the dendritic phase showed transition ISE behavior, while the interdendritic phase continued to display an inverse ISE. This phenomenon was observed for other high-entropy alloys, i.e., equiatomic MoNbTaVW and NbHfTiTaZr.

The relationship of the ISE between grain size in the polycrystalline matrix of pure Al and an Al–Cu alloy was studied Chinh et al. [47]. It is demonstrated that there is a close connection between the Hall–Petch relationship and the characteristics of the ISE phenomenon such that the ISE phenomenon may disappear in an ultrafine-grained matrix. This finding is significant in any attempts to interpret nanoindentation measurements performed on ultrafine-grained materials.

6. Conclusions

Aluminum (99.5%) was cold-deformed in compression tests with a degree of deformation up to 85% and a strain rate up to 2500 s⁻¹. Both the degree of deformation and the strain rate influenced microhardness and ISE parameters, above all, Meyer’s index n. The value of Meyer’s index for the non-deformed initial sample was close to 2. As the degree of deformation increased, its “reverse” character became more pronounced. Conversely, higher strain rates tended to shift the ISE into the “normal” range.

The deformation maps generated based on linear regression were less sensitive than the maps generated using Origin 8 and DoE.

Possible directions for future research are the use of more sensitive methods (measurement at low load) and comparison of the measured values with the calculated W, as well as measurements on other metallic and non-metallic materials (possibly also questioning the W parameter). In future research, we will also focus in more detail on the analysis of deformation inhomogeneity.